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Dirac Equation
The Dirac Equation is an attempt to make Quantum Mechanics Lorentz Invariant, i.e. incorporate Special Relativity. It attempted to solve the problems with the Klein-Gordon Equation. In Quantum Field Theory, it is the field equation for the spin-1/2 fields, also known as Dirac Fields. Statement \right)\Psi =0 }} Relationship with Klein-Gordon Equation The sense in which the Dirac Equation can be regarded as the square root of the Klein-Gordon Equation is made clear by the use of the Feynman Slash Notation.Particle Physics (3rd Edition), B. R. Martin, G.Shaw, Manchester Physics Series, John Wiley & Sons, ISBN 978-0-470-03294-7. The Klein-Gordon Equation can be factored: : 0 = (\hbar^2\partial^\mu \partial_\mu + (mc)^2)\psi = ((\hbar\partial\!\!\!/)^2 + (mc)^2)\psi = (i\hbar\partial\!\!\!/ + mc)(-i\hbar\partial\!\!\!/ + mc)\psi \,. The last factor, (-i\hbar\partial\!\!\!/ + mc)\psi \, , is simply the Dirac equation. Hence any solution to the Dirac Equation is automatically a solution to the Klein-Gordon Equation: : (-i\hbar\partial\!\!\!/ + mc)\psi = 0 \rightarrow (\hbar^2\partial^\mu \partial_\mu + (mc)^2)\psi = 0 \,. But the converse is not true; not all solutions to the Klein–Gordon equation solve the Dirac Equation. In a Potential In a potential, the Dirac Equationw takes the form: \left(i\hbar\not\nabla-mc_0\right)\psi=0 Where \nabla^\mu=\partial^\mu+ig A_\mu where g is the Coupling Constant. Free Particle Solution The Free Particle Solution takes the form: \psi = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{p}}}\sum_{s} \left( a^{s}_{\textbf{p}}u^{s}(p)e^{-ip \cdot x}+b^{s \dagger}_{\textbf{p}}v^{s}(p)e^{ip \cdot x}\right).\, Relationship with spin and the Pauli Theory Due to the implication of these complex scalar-valued Wavefunctions being replaced by spinor-valued fields, one finds that there are operators called spin operators. These spin operators' eigenvalues become the ordinary spin quantum numbers, and we see that they are quantised as \frac{n}{2} where n\in\mathbb Z , and does not exceed 4. The spins are therefore 0, \frac12, 1, \frac32, 2 . \ Note, that a \frac32 spin is possible only by Supersymmetry. This is in fact an argument in favour of Supersymmetry.' We furthermore see that bosons have integer spins, whereas fermions have half-integer spins. The actual spins can be shown to be representations of SO(n+1) or SO(n+\frac12) where n is the spin quantum number. Implications Antimatter It is seemingly paradoxical that the Dirac Equation allows for the existence of negative energies. This seems very strange at first glance. However, with resolute faith in Mathematics, one must accept the existence of these weird creatures. This became known as the model of Antimatter. Antimatter is matter with negative energy. The positron, for example, is a particle of antimatter, whereas the electron is one of matter. They both have the same mass, but the opposite charge. Spinors One sees that the solutions to the Dirac Equation take the form \psi^\mu , and become spinorial. Therefore, the Dirac Equation introduces Spinors into Physics. Quantum Field Theory As we have seen previously, the Dirac Equation acts indeed paradoxically if we continue to interpret the \psi as an ordinary wavefunction. Therefore, we must interpret it as a Field. This leads to the birth of Quantum Field Theory. Maximum Atomic Number for atoms In the Dirac Equation, one may calculate the eigenstates and find the eigenvalues of the Hamiltonian, one sees that the Energies of an electron 'bound to the nucleus are quantised as: E=mc_0^2\sqrt{1-\alpha^2Z^2} Here, Z is the number of electrons, \alpha is the Fine-Structure Constant, the coupling constant for Electromagnetism. If Z>137 , this becomes imaginary. Classically/(Purely Special Relativistically), this is interpreted as an Electron going faster than light, as can be seen from the apparent mass, energy, etc., of an object moving at superluminal speeds. Therefore, this means that the atom cannot have more than 137 electrons. For neutral atoms, this is equivalent to stating that the atomic number is never more than 137. For cations, however, it is possible to have a larger atomic number, since it has more protons than electrons. Category:Quantum Field Theory